I don't know how to prove the statement: every compact set is a finite union of disjoint connected compact subsets.
Here is my trial: If a compact subset A is connected, then we are done. Now, suppose that A is disconnected. Then there exists two compact subsets C1 and C2 of A which are open in A such that A=(C1)U(C2). If both of C1 and C2 are connected, then we are done. Suppose that at least one of C1 or C2 is disconnected. Then, w.l.o.g., let C2 be disconnected. Again, C2 can be separated by C3 and C4. Suppose that this process does not end in finite step. Then we have a disjoint open cover of A, so by the compactness of A, we have a finite subcover C1, ..., CN such that A=union of Ci from 1 to N. But then CN is disconnected from the supposition that the process does not end in finite step... Contradiction.
Does it work?
For a counterexample, consider the set $$\{0\}\cup\left\{1,{\small{\frac{1}{2}}},{\small{\frac{1}{3}}},...\right\}$$